Communications on Stochastic Analysis Serials Publications Vol. 10, No. 1 (2016) 57-81 www.serialspublications.com ON THE SECOND FUNDAMENTAL THEOREM OF ASSET PRICING RAJEEVA L. KARANDIKAR AND B. V. RAO Abstract. Let X1;:::;Xd be sigma-martingales on (Ω; F; P). We show that every bounded martingale (with respect to the underlying filtration) admits an integral representation with respect to X1;:::;Xd if and only if there is no equivalent probability measure (other than P) under which X1;:::;Xd are sigma-martingales. From this we deduce the second fundamental theorem of asset pricing- that completeness of a market is equivalent to uniqueness of Equivalent Sigma-Martingale Measure (ESMM). 1. Introduction The (first) fundamental theorem of asset pricing says that a market consisting of finitely many stocks satisfies the No Arbitrage (NA) property if and only there exists an Equivalent Martingale Measure (EMM)- i.e. there exists an equivalent probability measure under which the (discounted) stocks are (local) martingales. The No Arbitrage property has to be suitably defined when we are dealing in continuous time, where one rules out approximate arbitrage in the class of admis- sible strategies. For a precise statement in the case when the underlying processes are locally bounded, see Delbaen and Schachermayer [5]. Also see Bhatt and Karandikar [1] for an alternate formulation of a weaker result, where the approxi- mate arbitrage is defined only in terms of simple strategies. For the general case, the result is true when local martingale in the statement above is replaced by sigma-martingale. See Delbaen and Schachermayer [6]. They have an example where the No Arbitrage property holds but there is no equivalent measure under which the underlying process is a local martingale. However, there is an equivalent measure under which the process is a sigma-martingale. The second fundamental theorem of asset pricing says that the market is com- plete (i.e. every contingent claim can be replicated by trading on the underlying securities) if and only if the EMM is unique. Interestingly, this property was studied by probabilists well before the connection between finance and stochastic calculus was established (by Harrison{Pliska [9]). The completeness of market is same as the question: when is every martingale representable as a stochastic inte- gral with respect to a given set of martingales fM 1;:::;M dg. When M 1;:::;M d is the d-dimensional Wiener Process, this property was proven by Ito [10]. Jacod Received 2016-2-19; Communicated by the editors. 2010 Mathematics Subject Classification. Primary 60H05, 62P05; Secondary 60G44, 91B28. Key words and phrases. Equivalent martingale measure, martingale representation, sigma- martingales. 57 58 RAJEEVA L. KARANDIKAR AND B. V. RAO and Yor [13] proved that if M is a P-local martingale, then every martingale N admits a representation as a stochastic integral with respect to M if and only if there is no probability measure Q (other than P) such that Q is equivalent to P and M is a Q-local martingale. The situation in higher dimension is more com- plex. The obvious generalisation to higher dimension is not true as was noted by Jacod{Yor [13]. To remedy the situation, a notion of vector stochastic integral was introduced- where a vector valued predictable process is the integrand and vector valued mar- tingale is the integrator. The resulting integral yields a class larger than the linear space generated by component wise integrals. See [12], [2]. However, one has to prove various properties of the vector stochastic integrals once again. Here we achieve the same objective in another fashion avoiding defining integra- tion again from scratch. In the same breath, we also take into account the general case, when the underlying processes need not be bounded but satisfy the property NFLVR and thus one has an equivalent sigma-martingale measure (ESMM). To the best of our knowledge, the martingale representation property in the framework of sigma-martingales is not available in literature. Indeed, most treatments deal with square integrable martingales where the notion of orthogonality of martingales is available which simplifies the treatment. For a semimartingale X, let LR(X) denote the class of predictable processes f such that the stochastic integral f dX is defined. A semimartingale Z is said to admit an integral representation with respect to semimartingales (X1;X2;:::;Xd) if there exists a semimartingale Y and predictable processes f; gj such that f 2 L(Y ), gj 2 L(Xj) Z Xd t j j 8 ≥ Yt = Y0 + gs dXs ; t 0 j=1 0 and Z t Zt = Z0 + fs dYs; 8t ≥ 0: 0 In Theorem 5.3 we will show that for a multidimensional sigma-martingale (X1;X2;:::;Xd) all bounded martingales admit a representation with respect to Xj, 1 ≤ j ≤ d if and only if the ESMM is unique. The most critical part of its proof is to show that the class of martingales that admit representation with respect to (X1;X2;:::;Xd) is closed under L1 convergence : If M n are martingales such that there exist f n; gn;j, Y n with Z Xd t n n n;j j 8 ≥ Yt = Y0 + gs dXs ; t 0 j=1 0 and Z t n n n n 8 ≥ Mt = M0 + fs dYs ; t 0 0 j n − j ! and if E[ Mt Mt ] 0, then we need to show that M also admits a representa- tion with respect to (X1;X2;:::;Xd). When (X1;X2;:::;Xd) is d-dimensional Brownian motion, or when Xj and Xk are orthogonal as martingales, one can deduce that for each j, fgn;j : n ≥ 1g is Cauchy in an appropriate norm and SECOND FUNDAMENTAL THEOREM OF ASSET PRICING 59 thereby complete the proof. This step fails in general, as the Jacod{Yor example shows. So we need to do an orthogonalisation of (X1;X2;:::;Xd) to achieve the same. However, (X1;X2;:::;Xd) may not be square integrable and thus we need to change the measure to a measure Q (equivalent to the underlying probability measure P) to make it so. Under Q,(X1;X2;:::;Xd) need not be martingales. This part is delicately managed by keeping both P; Q in the picture. The rest of the argument is on the lines of Jacod{Yor [13]. When X1;X2;:::;Xd represent (prices of) stocks, Y can be thought of as (the price of) a mutual fund or an index fund and the investor is trading on such a fund trying to replicate the security Z. In this framework, Theorem 5.3 gives us the second fundamental theorem of as- set pricing : Market is complete if and only if ESMM (equivalent sigma-martingale measure) is unique. 2. Preliminaries and Notation Let us start with some notations. (Ω; F; P) denotes a complete probability space with a filtration (F) = fFt : t ≥ 0g such that F0 consists of all P-null sets (in F) and \t>sFt = Fs 8s ≥ 0: Thus, we can (and do) assume that all martingales have r.c.l.l. paths. For various notions, definitions and standard results on stochastic integrals, we refer the reader to Meyer [15], Jacod [11] or Protter [16]. Let M denote the class of martingales and Mloc denote the class of local mar- 2 M L1 tingales. For M loc, let m(M) be the class of predictable processes f such that there exists a sequence of stopping times σ " 1 with Z k σk 1 f 2 g 2 1 E[ fs d[M; M]s ] < : R 0 For such an f, N = f dM is defined and is a local martingale. For a semimartingale X, let L(X) denote the class of predictable process f such that X admits a decomposition X = N + A with N 2 Mloc , A being a process 2 L1 with finite variation paths with f m(N) and Z t jfsjdjAjs < 1 a:s: 8t < 1: (2.1) 0 R R R For f 2 L(X), the stochastic integral f dX is defined as f dN + f dA. It can be shown that the decomposition X = N + A is not unique, the definition does not depend upon the decomposition. See [11]. For M 1;M 2;:::;M d 2 M let C(M 1;M 2;:::;M d) denote the class of martin- 2 M 9 j 2 L1 j ≤ ≤ gales Z such that f m(M ); 1 j d with Z Xd t j j 8 ≥ Zt = Z0 + fs dMs ; t 0: j=1 0 For T < 1, let ~ 1 2 d 1 2 d KT (M ;M ;:::;M ) = fZT : Z 2 C(M ;M ;:::;M )g: 60 RAJEEVA L. KARANDIKAR AND B. V. RAO ~ For the case d = 1, Yor [18] had proved that KT is a closed subspace of 1 ~ 1 2 d L (Ω; F; P). The problem in case d > 1 is that in general KT (M ;M ;:::;M ) need not be closed. Jacod{Yor [13] gave an example where M 1;M 2 are continuous ~ 1 2 square integrable martingales and KT (M ;M ) is not closed. Jacod [12] defined the vector stochastic integral and Memin [14] proved that with the modified defi- nition of the integral, this space is closed. We will follow a different path. For martingales M 1;M 2;:::;M d, let F(M 1;:::;M d) be the class of martingales 2 M 9 2 C 1 d 2 L1 Z such that Y (M ;:::;M ) and f m(Y ) with Z t Zt = Z0 + fs dYs; 8t ≥ 0: 0 Let 1 2 d 1 2 d KT (M ;M ;:::;M ) = fZT : Z 2 F(M ;M ;:::;M )g: The main result of the next section is 1 2 d 1 2 d Theorem 2.1.